Elementary mathematics
Based on "A Calculus of Number Based on Spatial Forms" by Jeffrey M. James 1993.f1(x) = x f2(x) = -x f3(x) = ex f4(x) = ln x f3(f4(x)) = x eln x = x f4(f3(x)) = x ln ex = x <> - Inversion -0 = 0 () = o - Instanziation e0 = 1 [] = ∎ - Abstraction ln 0 = -∞ + [-∞...∞]i [(a)] = ([a]) = a Involution Axiom 1 ln ea = eln a = a (a[bc]) = (a[b])(a[c]) Distribution of a Axiom 2 ea + ln (b + c) = ea + ln b + ea + ln c a<a> = Inversion of a Axiom 3 a - a = 0 <<a>> = a Inverse Cancellation (double Inversion) Theorem 1 -(-a) = a <a><b> = <ab> Inverse Collection Theorem 2 (-a) + (-b) = -(a + b) Proof of Inverse Cancellation for a != ∎ <<a>> <<a>><a>a -Inversion of a a Inversion of <a> Proof of Inverse Collection for a,b != ∎ <a><b> <a><b>ab<ab> -Inversion of ab <ab> Inversion of a and b Proof of additive self-inversion of void (ambiguous sign) -0 = 0 <> Inversion of void Cardinality of a (second degree) (Count) a = ([a]) = ([a][o]) -Involution2, -Involution1 aa = ([a][o])([a][o]) = ([a][oo]) 2x a, -Distribution of [a] a...a = ([a][o...o]) induction step to all natural numbers a * b => ([a][b]) a / b => ([a]<[b]>) for b != Proof of Associativity of multiplication (a*b)*c = a*(b*c) ([([a] [b])][c] ) ( [a] [b] [c] ) Involution1 ( [a][([b] [c])]) -Involution1 Proof of multiplicative self-inversion of o 1/1 = 1 (<[o]>) (< >) Involution1 ( ) Inversion of void
Real part of the complex logarithm
Imaginary part of the complex logarithm - Riemann surface Absorption ([a]∎) = Zero cardinality of a Theorem 3 a * 0 = 0 ∎ a = ∎ Absorption of a (black hole) Theorem 4 ∎ + a = ∎ Proof of Zero cardinality ([a]∎) ([a]∎)([a][b])<([a][b])> -Inversion of ([a][b]) ([a][b])<([a][b])> -Distribution of [a] Inversion of ([a][b]) Proof of Absorption ∎ a [(∎ [(a)])] 2x -Involution1 [ ] Zero cardinality of (a) Proof of Zero cardinality ([a]∎) ( ∎) Absorption of [a] Involution2 Indefiniteness of cardinality of ∎ ∎ = ∎∎ = ∎∎∎ = ∎∎∎∎ = ... -Absorption of ∎ = <> = [o] = (∎) = [(<>)] = ([<>]) = <[o]> = <(∎)> 5x Inversion of void, 3x Involution1, 3x Involution2 Indefiniteness of Inversion of ∎ (<∎>) = ([o]<∎>) <= 1/0 is undefined (singularity) 0 * ? = 1 zero don't have multiplicative inverse 0/0 ambiguous 0/a = 0 a/a = 1 a/0 = ∞ inverse Absorption <∎> a <∎><<a>> -Inverse Cancellation <∎ <a>> Inverse Collection <∎ > Absorption of <a> ∎<∎> -Inversion of ∎ => Absorption of <∎> or inverse Absorption of ∎ => Possibility of arbitrary absorption of everything everywhere => <∎> undefined (Inversion of a black hole is not allowed) => Division by 0 is undefined (singularity) Cardinality of [a] (third degree) ([a]) = (([[a]])) = (([[a]][o])) -Involution2, -Involution1 ([a][a]) = (([[a]][o])([[a]][o])) = (([[a]][oo])) 2x [a], -Distribution of [[a]] ([a]...[a]) = (([[a]][o...o])) induction step to all natural numbers a b Addition ooo ooo 3 + 3 = 6 ( [a] [b]) Multiplication ( [ooo] [ooo]) 3 * 3 = 3 + 3 + 3 = 9 (( [[a]] [b])) Exponentiation (( [[ooo]] [ooo])) 33 = 3 * 3 * 3 = 27 ((( [[[a]]] [b]))) Tetration ((( [[[ooo]]] [ooo]))) 333 = 327 = 7.625.597.484.987 (((([[[[a]]]][b])))) Hyper-5 (((([[[[ooo]]]][ooo])))) 3...3 7.625.597.484.987 times - our universe is already too small for that ... ab => (([[a]][b])) a0 = 1 => (([[a]]∎)) = ((∎)) = o Absorption of [[a]], Involution2 a1 = a => (([[a]][o])) = (([[a]])) = ([a]) = a Involution1, 2x Involution2 Inversion ([<a>]b) = <([a]b)> Inverse Promotion1 Theorem 5 -a * eb = -(a * eb) (<[<a>]>) = <(<[a]>)> Inverse Promotion2 Axiom 4 1/-a = -1/a (<[<a>]>b) = <(<[a]>b)> Inverse Promotion3 Theorem 6 1/-a * eb = -(1/a * eb) Proof of Inverse Promotion1 for a != ∎ ([<a>]b) <([a]b)>([a]b)([<a>]b) -Inversion of ([a]b) <([a]b)>([a <a>]b) -Distribution von b <([a]b)>([ ]b) Inversion of a <([a]b)>([ ] ) Absorption von b <([a]b)> Involution2 Proof of Inverse Promotion3 for a != ∎ ( <[<a>]> b) <(<[a]>b)>( <[a]> b)( <[<a>]> b) -Inversion of (<[a]>b) <(<[a]>b)>([(<[a]>)]b)([ (<[<a>]>) ]b) 2x -Involution1 <(<[a]>b)>([(<[a]>) (<[<a>]>) ]b) -Distribution von b <(<[a]>b)>([(<[a]>) <(<[ a ]>)>]b) Inverse Promotion2 <(<[a]>b)>([ ]b) Inversion of (<[a]>) <(<[a]>b)>([ ] ) Absorption von b <(<[a]>b)> Involution2 Proof of irrationality of √2 √2 = a/b 2 = (a/b)2 2 * b2 = (a/b)2 * b2 2 * b * b = a * a a mod 1 = 0 & b mod 1 = 0 -> -> (a*a) mod 2 = 0 -> a mod 2 = 0 -> (2*b*b) mod 4 = 0 -> (b*b) mod 2 = 0 -> b mod 2 = 0 -> (a*a) mod 8 = 0 -> a mod 4 = 0 -> (2*b*b) mod 16 = 0 -> (b*b) mod 8 = 0 -> b mod 4 = 0 -> (a*a) mod 32 = 0 -> a mod 8 = 0 -> (2*b*b) mod 64 = 0 -> (b*b) mod 32 = 0 -> b mod 8 = 0 -> (a*a) mod 128 = 0 -> ... => √2 is irrational (([[oo]]<[oo]>)) = ([a]<[b]>) ([[oo]]<[oo]>) = [a]<[b]> b = o -> a = (([[oo]]<[ oo ]>)) eln 2 / 2 a = o -> b = (([[oo]]<[<oo>]>)) eln 2 / -2 b = oo -> a = (([[oo]]<[ oo ]>)[oo]) eln 2 / 2 + ln 2 a = oo -> b = (([[oo]]<[<oo>]>)[oo]) eln 2 / -2 + ln 2 b = ooo -> a = (([[oo]]<[ oo ]>)[ooo]) eln 2 / 2 + ln 3 a = ooo -> b = (([[oo]]<[<oo>]>)[ooo]) eln 2 / -2 + ln 3 ... => a and b are not naturally resolvable together => (([[oo]]<[oo]>)) is irrational Operators -a => <a> for a != ∎ a+b => ab a+b+c => abc a-b => a<b> for b != ∎ a*b => ([a][b]) a*b*c => ([a][b][c]) a*-b => ([a][<b>]) = <([a][b])> = ([<a>][b]) +-Inverse Promotion1 for a,b != ∎ a/b => ([a]<[b]>) for b != 1/b => ([o]<[b]>) = (<[b]>) Involution1 for b != a/a => ([a]<[a]>) = o Inversion of [a] for a != ab => (([[a]][b])) a1 => (([[a]][o])) = (([[a]])) = a Involution1, 2x Involution2 a-b => (([[a]][<b>])) for b != ∎ a1/b => (([[a]]<[b]>)) = (([[a]][(<[b]>)])) -Involution1 for b != ac/b => (([[a]]<[b]>[c])) = (([[a]][(<[b]>[c])])) -Involution1 for b != abc (([[a]][(([[b]][c]))])) (([[a]] ([[b]][c]) )) Involution1 ((ab)c)1/d = ab*c/d (([[(([[(([[a]][b]))]][c]))]]<[d]>)) (( [[a]][b] [c] <[d]>)) 4x Involution1 ab * ac = ab+c ([(([[a]][b]))][(([[a]][c]))]) ( ([[a]][b]) ([[a]][c]) ) 2x Involution1 ( ([[a]][b c]) ) Distribution of [[a]] ac * bc = (a*b)c ([(([[a]][c]))][(([[b]][c]))]) ( ([[a]][c]) ([[b]][c]) ) 2x Involution1 ( ([[a] [b]][c]) ) Distribution of [c] 1 / e = e-1 => ([o]<[(o)]>) = (<[(o)]>) = (<o>) 2x Involution1 1 / e2 = e-2 => ([o]<[(oo)]>) = (<[(oo)]>) = (<oo>) 2x Involution1 ln (a*b) = ln a + ln b => [([a][b])] = [a][b] Involution1 ln (a/b) = ln a - ln b => [([a]<[b]>)] = [a]<[b]> Involution1 ln (ab) = ln a * b => [(([[a]][b]))] = ([[a]][b]) Involution1 logb a = ln a/ln b => ([[a]]<[[b]]>) 1/(1/a) = a (<[(<[a]>)]>) (< <[a]> >) Involution1 ( [a] ) Inverse Cancellation a Involution2 (1/a)*(1/b) = 1/(a*b) ([(< [a]>)][(<[b] >)]) ( < [a]> <[b] > ) 2x Involution1 ( < [a] [b] > ) Inverse Collection ( <[([a] [b])]> ) -Involution1 (1/a)b = a-b ( ([[(<[a]>)]][ b ]) ) ( ([ <[a]> ][ b ]) ) Involution1 (<([ [a] ][ b ])>) Inverse Promotion1 ( ([ [a] ][<b>]) ) -Inverse Promotion1 a/c + b/d = (a*d + b*c) / c*d ( [a] <[c]>)( [b] <[d]>) ( [a][d] <[d]><[c]>)( [b][c] <[c]><[d]>) -Inversion of [d] and [c] ( [a][d] <[d] [c]>)( [b][c] <[c] [d]>) 2x Inverse Collection ([([a][d])]<[d] [c]>)([([b][c])]<[c] [d]>) 2x -Involution1 ([([a][d]) ([b][c])]<[c] [d]>) -Distribution of <[c][d]> 1/a + 1/b = (a + b) / a*b ( <[a]>)( <[b]>) ([b]<[b]><[a]>)([a]<[a]><[b]>) -Inversion of [b] and [a] ([b]<[b] [a]>)([a]<[a] [b]>) 2x Inverse Collection ([ab]<[a][b]>) -Distribution of <[a][b]> (a + b) * (a - b) = a² - b² ([a b][a <b>]) ([a b][a]) ([a b][<b>]) Distribution of [ab] ([a][a])([b][a]) ([a][<b>]) ([b][<b>]) Distribution of [a] and [<b>] ([a][a])([b][a])<([a][ b ])><([b][ b ])> 2x -Inverse Promotion1 ([a][a]) <([b][ b ])> Inversion of ([a][b]) (([[a]][oo])) <(([[b]][oo]))> Cardinality of [a] and [b] Numbers 0 => 1 => o 2 => oo -1 => <o> -2 => <oo> 1/2 => (<[oo]>) 2/3 => ([oo]<[ooo]>) 43 => ([b][oooo])ooo for b = oooooooooo 243 => ([b][([b][oo])oooo])ooo for b = oooooooooo 1243 => ([b][([b][boo])oooo])ooo for b = oooooooooo Arity {a} = ([oooooooooo][a]) Definition of {} oooooooooo{a} = {o a} Carry+ Cardinality of oooooooooo {a}{b} = {ab} Collection+ ([{a}]b) = {([a]b)} Promotion+ {} = Zero cardinality+ 23 * 114 = 2622 ([ {oo}ooo ] [ {{o}o}oooo ]) ([{oo}][{{o}o}oooo]) ([ooo][{{o}o}oooo]) Distribution of [{{o}o}oooo] {([ oo ][{{o}o}oooo])} ([ooo][{{o}o}oooo]) Promotion+ {{{o}o}oooo{{o}o}oooo} {{o}o}oooo{{o}o}oooo{{o}o}oooo 2x -Cardinality of {{o}o}oooo {{{o}o}oooo{{o}o}oooo} {{o}o}{{o}o}{{o}o}oooooooooooo Collection+ of o {{{o}o}oooo{{o}o}oooo} {{o}o}{{o}o}{{o}oo} oo Carry+ {{{o}o} {{o}o} {o}{o}{o} oooooooooooo} oo Collection+ of {o} {{{o}o} {{o}o} {o}{o}{oo} oo} oo Carry+ {{{o} {o} oooooo} oo} oo Collection+ of {{o}} {{{ oo} oooooo} oo} oo Collection+ of {{{o}}} Inverse Arity /a\ = (<[oooooooooo]>[a]) Definition of /\ /oooooooooo a\ = o/a\ Carry- Cardinality of oooooooooo /a\/b\ = /ab\ Collection- ([/a\]b) = /([a]b)\ Promotion- /\ = Zero cardinality- {/a\} = /{a}\ = a Arity Cancellation 12,1 * 1,012 = 12,2452 ([ {o}oo/o\ ] [ o//o/oo\\\ ]) ([{o}][o//o/oo\\\]) ([oo][o//o/oo\\\]) ([/o\][o//o/oo\\\]) 2x Distribution of [o//o/oo\\\] {([ o ][o//o/oo\\\])}([oo][o//o/oo\\\])/([ o ][o//o/oo\\\])\ 2x Promotion ± { o//o/oo\\\ } o//o/oo\\\o//o/oo\\\ /o//o/oo\\\ \ 3x -Cardinality of o//o/oo\\\ {o}{ //o/oo\\\ } o//o/oo\\\o//o/oo\\\ /o//o/oo\\\ \ -Collection+ {o} /o/oo\\ o//o/oo\\\o//o/oo\\\ /o//o/oo\\\ \ Arity Cancellation1 {o}oo /o/oo\\ //o/oo\\\ //o/oo\\\ /o//o/oo\\\ \ Collection+ of o {o}oo /oo/oo\ /o/oo\\ /o/oo\\ //o/oo\\\ \ Collection- of /o\ {o}oo /oo/oooo /oo\ /oo\ /o/oo\\\ \ Collection- of //o\\ {o}oo /oo/oooo /ooooo /oo\\\ \ Collection- of ///o\\\ 100 / 3 = 33,3... ([ {{o}} ]<[ooo]>) {([ {o} ]<[ooo]>)} Promotion+ {([oooooooooo]<[ooo]>)} -Carry+ {([ooo]<[ooo]>)([ooo]<[ooo]>)([ooo]<[ooo]>) ([o]<[ooo]>)} 3x Distribution of <[ooo]> {( )( )( ) ([o]<[ooo]>)} 3x Inversion of [ooo] {ooo} {([o]<[ooo]>)} -Collection+ {ooo} ([{o}]<[ooo]>) -Promotion+ {ooo} ([oooooooooo]<[ooo]>) -Carry+ {ooo}([ooo]<[ooo]>)([ooo]<[ooo]>)([ooo]<[ooo]>) ([o]<[ooo]>) 3x Distribution of <[ooo]> {ooo}( )( )( ) ([o]<[ooo]>) 3x Inversion of [ooo] {ooo}ooo ([/oooooooooo\]<[ooo]>) -Carry- {ooo}ooo /([ oooooooooo ]<[ooo]>) \ Promotion- {ooo}ooo/([ooo]<[ooo]>)([ooo]<[ooo]>)([ooo]<[ooo]>)([o]<[ooo]>) \ 3x Distribution of <[ooo]> {ooo}ooo/( )( )( )([o]<[ooo]>) \ 3x Inversion of [ooo] {ooo}ooo/ooo ([/oooooooooo\]<[ooo]>) \ -Carry- {ooo}ooo/ooo /([ oooooooooo ]<[ooo]>)\\ Promotion- ... 100 / 3,3 = 30,3030... ([ {{o}} ]<[ooo/ooo\]>) {([ {o} ]<[ooo/ooo\]>)} Promotion+ {([oooooooooo]<[ooo/ooo\]>)} -Carry+ {([ooooooooo/ooo\/ooo\/ooo\/o\]<[ooo/ooo\]>)} -Carry- and 3x -Collection- {([ooo/ooo\]<[ooo/ooo\]>)(dito)(dito) ([ /o\ ]<[ooo/ooo\]>)} 3x Distribution of <[ooo/ooo\]> {( )( )( ) ([ /o\ ]<[ooo/ooo\]>)} 3x Inversion of [ooo/ooo\] {ooo} {([ /o\ ]<[ooo/ooo\]>)} -Collection+ {ooo} ([{/o\}]<[ooo/ooo\]>) -Promotion+ {ooo} ([ o ]<[ooo/ooo\]>) Arity Cancellation1 {ooo} ([/oooooooooo\]<[ooo/ooo\]>) -Carry- {ooo} /([ oooooooooo ]<[ooo/ooo\]>) \ Promotion- {ooo} /([ooooooooo/ooo\/ooo\/ooo\/o\]<[ooo/ooo\]>) \ -Carry- and 3x -Collection- {ooo}/([ooo/ooo\]<[ooo/ooo\]>)(dito)(dito) ([/o\]<[ooo/ooo\]>) \ 3x Distribution of <[ooo/ooo\]> {ooo}/( )( )( ) ([/o\]<[ooo/ooo\]>) \ 3x Inversion of [ooo/ooo\] {ooo}/ooo ([//oooooooooo\\]<[ooo/ooo\]>) \ -Carry- {ooo}/ooo //([ oooooooooo ]<[ooo/ooo\]>) \\\ 2x Promotion- {ooo}/ooo //([ooooooooo/ooo\/ooo\/ooo\/o\]<[ooo/ooo\]>) \\\ -Carry- and 3x -Collection- {ooo}/ooo//([ooo/ooo\]<[ooo/ooo\]>)(dito)(dito)([/o\]<[ooo/ooo\]>) \\\ 3x Distribution of <[ooo/ooo\]> {ooo}/ooo//( )( )( )([/o\]<[ooo/ooo\]>) \\\ 3x Inversion of [ooo/ooo\] {ooo}/ooo//ooo ([//oooooooooo\\]<[ooo/ooo\]>) \\\ -Carry- {ooo}/ooo//ooo //([ oooooooooo ]<[ooo/ooo\]>)\\\\\ 2x Promotion- ...
Real part of the complex exponential function
Imaginary part of the complex exponential function Transcendental J = [<o>] Phase Element Definition ln -1 = πi (~3.1415926535i) (J) = ([<o>]) = <o> Involution2 eJ = eπi = -1 (o) Euler number e1 = e (~2.71828182845) ((o)) ee (~15.1542622414) [<(a)>] = a[<o>] Phase Independence of a Theorem 7 ln -ea = a + ln -1 (a JJ ) = (a) J-cancellation1 Theorem 8 ea + 2*ln -1 = ea + ln (-1)² = ea + ln 1 = ea = ea + 2πi (a<bJJ>) = (a<b>) J-cancellation2 Theorem 9 ea - b - 2*ln -1 = ea - b - ln (-1)² = ea - b - ln 1 = ea - b = ea - b - 2πi (J) = (<J>) Self-Inversion of J Theorem 10 eJ = e-J (([J]<[oo]>)) i and <i> eJ/2 = √-1 = ±i i = (<[<i>]>) = <(<[i]>)> imaginary unit Theorem 11 i = 1/-i = -e-ln i ([<J>][i]) = ([J][<i>]) Circle number Pi -J*i = J/i = J*-i = π (~3.1415926535) Like ln in the imaginary, by an arbitrary multiple of 2π is ambiguous, so at e all values are repeated every 2πi. To represent this, the J-cancellation and self-inversion of J is permissible only within an instantiation, or they must be contained in at least one more instantiation than in abstractions. Proof of Phase Independence of a for a != [∎] [<(a)> ] [<(a)>( [ ])] -Involution2 [<(a)>(a[ ])] -Absorption of a [<(a)>(a[o <o>])] -Inversion of o [<(a)>(a[o])(a[<o>])] Distribution of a [<(a)>(a )(a[<o>])] Involution1 [ (a[<o>])] Inversion of (a) a[<o>] Involution1 Proof of J-cancellation within () [<o>][<o>] [<([<o>])>] -Phase Independence von [<o>] [< <o> >] Involution2 [ o ] Inverse Cancellation Involution1 Proof of J-cancellation1 within () (a[<o>][<o>]) -Involution1 <<(a[ o ][ o ])>> 2x Inverse Promotion1 (a[ o ][ o ]) Inverse Cancellation (a ) 2x Involution1 Proof of Self-Inversion of J within () (J ) (J< >) -Inversion of void (J<J J>) -J-cancellation2 (J<J><J>) -Inverse Collection ( <J>) Inversion of J
f1(x) = x2 f2(x) = √x f1(f2(x)) = x (√x)2 = x f2(f1(x)) = ±x √(x2) = ±x
Real part and imaginary part of the complex square root
Real part and imaginary part of the complex cube root Proof of ambiguity of (([J]<[oo]>)) or √-1 Whenever the exponent is a non-integer rational number, there is ambiguity. And if the exponent is less than or equal to 0, the base equals 0 gives a degenerate point. i * i = -i * -i = (±i)2 = -1 ([(([J]<[oo]>))][(([J]<[oo]>))]) (([[(([J]<[oo]>))]][oo])) Cardinality of [(([J]<[oo]>))] (( [J]<[oo]> [oo])) 2x Involution1 (( [J] )) Inversion of [oo] <o> 2x Involution2 and ([<(([J]<[oo]>))>][<(([J]<[oo]>))>]) <<([ (([J]<[oo]>)) ][ (([J]<[oo]>)) ])>> 2x Inverse Promotion1 ([ (([J]<[oo]>)) ][ (([J]<[oo]>)) ]) Inverse Cancellation (([[(([J]<[oo]>))]][oo]) ) Cardinality of [(([J]<[oo]>))] (( [J]<[oo]> [oo]) ) 2x Involution1 (( [J] ) ) Inversion of [oo] <o> 2x Involution2 Proof of imaginary unit1 (<[<<(([ J ]<[oo]>))>>]>) (< ([ J ]<[oo]>) >) Inverse Cancellation, Involution1 ( ([<J>]<[oo]>) ) -Inverse Promotion1 ( ([ J ]<[oo]>) ) -Self-Inversion of J Proof of imaginary unit2 <(<[(([ J ]<[oo]>))]>)> <(< ([ J ]<[oo]>) >)> Involution1 <( ([<J>]<[oo]>) )> -Inverse Promotion1 <( ([ J ]<[oo]>) )> -Self-Inversion of J -1 * -1 = 1 ([<o>][<o>]) ( ) J-cancellation1 ee = ee (([[(o)]][(o)])) (( o )) 3x Involution1 < a > = ([<([a])>]) = ([a][<o>]) = ([a]J) 2x-Involution2, Phase Independence of [a] [< a >] = [<([a])>] = [a][<o>] = [a]J -Involution2, Phase Independence of [a] <( a )> = ([<( a )>]) = (a[<o>]) = (aJ) -Involution2, Phase Independence von a Indefiniteness of Cardinality of J within () ... = (<JJJJ>) = (<JJ>) = o = (JJ) = (JJJJ) = ... J-cancellation1, J-cancellation2, Inversion of void ... = (<JJJ>) = (<J>) = (J) = (JJJ) = (JJJJJ) = ... J-cancellation1, J-cancellation2, Self-Inversion of J Indefiniteness of [∎] within () ( [∎] ) ( [∎] JJ) -J-cancellation1 ([<([∎])>]J) -Phase Independence of [∎] ([< ∎ >]J) Involution2 - Indefiniteness der Inversion of ∎ Indefiniteness of 00 (([∎]∎)) (( ∎)) Absorption of [∎] ( ) Involution2 => 00 = 1 but from the Indefiniteness of [∎] follows in principle the Indefiniteness of 00 a0 = 1 (([[a]]∎)) (( ∎)) Absorption of [[a]] ( ) Involution2 0n = 0 n ∈ ℕ (([∎][o...o])) (∎...∎) Cardinality of ∎ (∎ ) Absorption of ∎ Involution2 ln 0 + ln -1 = ln 0 ∎J [<(∎)>] -Phase Independence of ∎ [< >] Involution2 [ ] Inversion of void Proof of multiplicative Self-Inversion of (([J]<[oo]>)) (like o) 1/±i = ∓i (<[(([ J ]<[oo]>))]>) (< ([ J ]<[oo]>) >) Involution1 ( ([<J>]<[oo]>) ) -Inverse Promotion1 ( ([ J ]<[oo]>) ) -Self-Inversion of J and (<[<(([ J ]<[oo]>))>]>) <(<[ (([ J ]<[oo]>)) ]>)> -Inverse Promotion2 <(< ([ J ]<[oo]>) >)> Involution1 <( ([<J>]<[oo]>) )> -Inverse Promotion1 <( ([ J ]<[oo]>) )> -Self-Inversion of J ∓i = ±i <(([ J ]<[oo]>))> (<[<<(([ J ]<[oo]>))>>]>) imaginary unit11 (< ([ J ]<[oo]>) >) Inverse Cancellation, Involution1 ( ([<J>]<[oo]>) ) -Inverse Promotion1 ( ([ J ]<[oo]>) ) -Self-Inversion of J ±i = ∓i (([ J ]<[oo]>)) <(<[(([ J ]<[oo]>))]>)> imaginary unit12 <(< ([ J ]<[oo]>) >)> Involution1 <( ([<J>]<[oo]>) )> -Inverse Promotion1 <( ([ J ]<[oo]>) )> -Self-Inversion of J J/i = i*π / i = π ([J]<[(([ J ]<[oo]>))]>) ([J]< ([ J ]<[oo]>) >) Involution1 ([J] ([<J>]<[oo]>) ) -Inverse Promotion1 ([J] ([ J ]<[oo]>) ) -Self-Inversion of J and ([ J ]<[<(([ J ]<[oo]>))>]>) <([ J ]<[ (([ J ]<[oo]>)) ]>)> Inverse Promotion3 ([<J>]<[ (([ J ]<[oo]>)) ]>) -Inverse Promotion1 ([<J>]< ([ J ]<[oo]>) >) Involution1 ([<J>] ([<J>]<[oo]>) ) -Inverse Promotion1 ([<J>] ([ J ]<[oo]>) ) -Self-Inversion of J J * -i = π ([ J ][<(([ J ]<[oo]>))>]) <([ J ][ (([ J ]<[oo]>)) ])> Inverse Promotion1 ([<J>][ (([ J ]<[oo]>)) ]) -Inverse Promotion1 ([<J>] ([ J ]<[oo]>) ) Involution1 and ([J][<<(([ J ]<[oo]>))>>]) ([J][ (([ J ]<[oo]>)) ]) Inverse Cancellation ([J] ([ J ]<[oo]>) ) Involution1 π * i = J ([([<J>][i])][i]) ( [<J>][i] [i]) Involution1 ( [<J>][<o> ]) i * i = -1 <<( [ J ][ o ])>> 2x Inverse Promotion1 J Inverse Cancellation, Involution1, 2x Involution2 ii = eJ*i/2 = e-π/2 = e-π/2-2πk k ∈ ℤ (([[(([J]<[oo]>))]][(([J]<[oo]>))])) (( [J]<[oo]> ([J]<[oo]>) )) 3x Involution1 and ( ([[(([ J ]<[oo]>))]][<(([J]<[oo]>))>]) ) (<( [ J ]<[oo]> [ (([J]<[oo]>)) ])>) 2x Involution1, Inverse Promotion1 ( ( [<J>]<[oo]> [ (([J]<[oo]>)) ]) ) -Inverse Promotion1 ( ( [ J ]<[oo]> ([J]<[oo]>) ) ) Involution1, -Self-Inversion of J i-2i = eπ (([[(([J]<[oo]>))]][(([J]<[oo]>))][oo])) (( [J]<[oo]> ([J]<[oo]>) [oo])) 3x Involution1 (( [J] ([J]<[oo]>) )) Inversion of [oo] and ( ([[(([ J ]<[oo]>))]][<(([J]<[oo]>))>][oo]) ) (<( [ J ]<[oo]> [ (([J]<[oo]>)) ][oo])>) 2x Involution1, Inverse Promotion1 ( ( [<J>]<[oo]> [ (([J]<[oo]>)) ][oo]) ) -Inverse Promotion1 ( ( [ J ] ([J]<[oo]>) ) ) Involution1, -Self-Inversion of J 1/√a = √(1/a) (<[(([ [a] ]<[oo]>))]>) (< ([ [a] ]<[oo]>) >) Involution1 ( ([<[a]>]<[oo]>) ) -Inverse Promotion1 √9 = √±32 = ±3 (([[ ooooooooo ]]<[oo]>)) (([[( [ooo][ooo] )]]<[oo]>)) Cardinality of ooo (([[(([[ooo]][oo]))]]<[oo]>)) Cardinality of [ooo] (( [[ooo]][oo] <[oo]>)) 2x Involution1 (( [[ooo]] )) Inversion of [oo] ooo 2x Involution2 and (([[ ooooooooo ]]<[oo]>)) (([[ ( [ ooo ][ ooo ]) ]]<[oo]>)) Cardinality of ooo (([[<<( [ ooo ][ ooo ])>>]]<[oo]>)) -Inverse Cancellation (([[ ( [<ooo>][<ooo>]) ]]<[oo]>)) 2x -Inverse Promotion1 (([[ (([[<ooo>]][oo]) ) ]]<[oo]>)) Cardinality of [<ooo>] (( [[<ooo>]][oo] <[oo]>)) 2x Involution1 (( [[<ooo>]] )) Inversion of [oo] <ooo> 2x Involution2 -22 = 22 (([[<oo>]][oo])) ([<oo>][<oo>]) -Cardinality of [<oo>] <<([ oo ][ oo ])>> 2x Inverse Promotion1 ([ oo ][ oo ]) Inverse Cancellation (([[oo]][oo])) Cardinality of [oo] ab = -ab (( [[ a ]] [b])) (( [[ a ]][oo] <[oo]>[b])) -Inversion of [oo] (([[ (([[ a ]][oo])) ] ]<[oo]>[b])) 2x -Involution1 (([[ ( [ a ][ a ] ) ] ]<[oo]>[b])) -Cardinality of [a] (([[<<( [ a ][ a ] )>>] ]<[oo]>[b])) -Inverse Cancellation (([[ ( [<a>][<a>] ) ] ]<[oo]>[b])) 2x -Inverse Promotion (([[ (([[<a>]][oo])) ] ]<[oo]>[b])) Cardinality of [<a>] (( [[<a>]][oo] <[oo]>[b])) 2x Involution1 !!! ambiguity (( [[<a>]] [b])) Inversion of [oo] Self-Inversion problem -0 = 0 1/1 = 1 <∎> ∎ = ∎∎ = ∎∎∎ = ∎∎∎∎ = ... J = <J> ... = <JJJJ> = <JJ> = = JJ = JJJJ = ... , ambiguity of (([J]<[oo]>)) 1/±i = ∓i ab = -ab
Three-dimensional representation of the Eulerian formula ez*i = cos z + i sin z => (([z]([J]<[oo]>)))
Real part of the complex sine function
Imaginary part of the complex sine function an+1 = ([an]J) o -> (J) -> o -> (J) -> o -> ... <= (1 -> -1 -> 1 -> -1 -> 1 -> ...) an+1 = ([an][i]) o -> i -> (J) -> <i> -> o -> ... <= (1 -> i -> -1 -> -i -> 1 -> ...) an+1 = <([an][i])> o -> <i> -> (J) -> i -> o -> ... <= (1 -> -i -> -1 -> i -> 1 -> ...) ea*i = cos a + i sin a => (([a][i])) e-a*i = cos a - i sin a => (([<a>][i])) sin a = (ea*i - e-a*i)/2i => ([ (([a][i]))<(([<a>][i]))>]<[oo][i]>) cos a = (ea*i + e-a*i)/2 => ([ (([a][i])) (([<a>][i])) ]<[oo] >) ea = cosh a + sinh a => (a) e-a = cosh a - sinh a => (<a>) sinh a = (ea - e-a)/2 => ([(a)<(<a>)>]<[oo]>) cosh a = (ea + e-a)/2 => ([(a) (<a>) ]<[oo]>) sin(x + i y) = sin(x) cosh(y) + i cos(x) sinh(y) cos(x + i y) = cos(x) cosh(y) - i sin(x) sinh(y) sin(z) = -i sinh(i z) sinh(z) = -i sin(i z) cos(z) = cosh(i z) cosh(z) = cos(i z) sin'(z) = cos(z) sinh'(z) = cosh(z) cos'(z) = -sin(z) cosh'(z) = sinh(z) unit circle x2 + y2 = 1 (([[x]][oo])) (([[y]][oo])) = o unit hyperbole x2 - y2 = 1 (([[x]][oo]))<(([[y]][oo]))> = o
